Master equation approach to the n-coalescent problem

Given an evolutionary model, such as Wright-Fisher or Moran, the n-coalescent problem consists of going backward in time to find, for example, the time to the most recent common ancestor (MRCA) and the topology of the tree. In the literature, this problem is mainly addressed by directly computing the random variable t, the time to reach the MRCA. It is shown here that by shifting the focus from the random variable t to the joined variable (n,t), where n is the number of ancestors at time t, the problem is greatly simplified. Indeed, P(n,t), the probability of this variable, obeys a simpler master equation that can be solved in a straightforward way for the most general model. This probability can then be used to compute relevant information of the ncoalescent for both random variables t_{n} (random time to reach a given state n) and n_{t} (random number of ancestors at a given time t). The cumulative distribution function for t_{1}, for example, is P(1,t). This article presents a unified framework for coalescent models-whether time continuous or time discrete-along with their explicit analytical solutions, expressed in terms of transition rates between states. The application of this method to the Moran model or the Kingman's coalescent, where the results are known, is provided for comparison.